CVA_PM

Aim: Dimension reduction technique that maximises variation between classes while minimising within class variation.

This is achieved by the following tasks:

• Decomposing Variance
• Find a linear mapping to canonical space.
• Find a low dimensional approximation

The classical variance decomposition \[\mathbf{T}=\mathbf{B}+\mathbf{W},\]

has as an analogy in this setting \[ \mathbf{X'X} = \mathbf{\bar{\mathbf{X}}'C \bar{\mathbf{X}}} + \mathbf{X' [I - G(G'G)^{-1}C(G'G)^{-1}G'] X}. \]

The choice of \(\mathbf{C}\) determines the variant of CVA:

• Weighted: \(\mathbf{C}=\mathbf{N}=\mathbf{G'G}\)
• Unweighted: \(\mathbf{C}=\mathbf{I}_G - G^{-1}\mathbf{1}_G\mathbf{1}_G'\)
• Unweighted \w weighted centroid: \(\mathbf{C}=\mathbf{I}_G\)

Find a linear mapping

\[\mathbf{Y}=\mathbf{X}\mathbf{M}, \tag{1}\]

such that \[\frac{\mathbf{m}'\mathbf{B}\mathbf{m}}{\mathbf{m}'\mathbf{W}\mathbf{m}} \tag{2}\] is maximised s.t. \(\mathbf{m}'\mathbf{W}\mathbf{m}=1\).

It can be shown that this leads to the following equivalent eigen equations:

\[ \mathbf{W}^{-1}\mathbf{BM} = \mathbf{M \Lambda} \tag{3} \]

\[ \mathbf{BM} = \mathbf{WM \Lambda} \tag{4} \]

\[ (\mathbf{W}^{-\frac{1}{2}} \mathbf{B} \mathbf{W}^{-\frac{1}{2}}) \mathbf{M} = (\mathbf{W}^{-\frac{1}{2}} \mathbf{M}) \mathbf{\Lambda} \tag{5} \]

with \(\mathbf{M'BM}= \mathbf{\Lambda}\) and \(\mathbf{M'WM}= \mathbf{I}\).

Since the matrix \(\mathbf{W}^{-\frac{1}{2}} \mathbf{B} \mathbf{W}^{-\frac{1}{2}}\) is symmetric and positive semi-definite the eigenvalues in the matrix \(\mathbf{\Lambda}\) are positive and ordered. The rank of \(\mathbf{B} = min(p, G-1)\) so that only the first \(rank(\mathbf{B})\) eigenvalues are non-zero. We form the canonical variates with the transformation

\[ \bar{\mathbf{Y}} = \bar{\mathbf{X}}\mathbf{M}.\tag{5} \]

The first two canonical variates are given by:

\[\mathbf{\bar{Z}}=\mathbf{\bar{Y}}\mathbf{J}_2=\mathbf{\bar{X}}\mathbf{M}\mathbf{J}_2 \tag{6}\] where \(\mathbf{J'}_2=[\mathbf{I}_2 \quad \mathbf{0}]\). We add the individual sample points with the same transformation \[\mathbf{Z}=\mathbf{X}\mathbf{M}\mathbf{J}_2. \tag{7}\]

A new sample point, \(\mathbf{x}^*\), can be added by interpolation \[\mathbf{z}^*=\mathbf{x}^*\mathbf{M}\mathbf{J}_2.\tag{8}\]

Contains the following information on how well the biplot represents the information of the original and canonical space:

• quality: Quality of fit for canonical and original variables
• adequacy: Adequacy of original variables
• axis.predictivity: Axis predictivity
• within.class.axis.predictivity: Class predictivity
• within.class.sample.predictivity: Sample predictivity

The summary() function prints to screen the fit.measures stored in the object of class biplot.

The rotate() function rotates the samples and axes in the biplot by rotate.degrees degrees.

The reflect() function reflects the samples and axes in the biplot along an axis, x(horisontal reflection), y (vertical reflection) or xy (diagonal reflection).

The argument zoom= is FALSE by default. If zoom=TRUE a new graphical device is launched. The user is prompted to click on the desired upper left hand and lower right hand corners of the zoomed in plot.